Re: Does the number of nines increase?

Am 09.07.2024 um 22:07 schrieb Chris M. Thomasson:

On 7/9/2024 5:11 AM, WM wrote:

Le 09/07/2024 à 03:04, Moebius a écrit :

Am 07.07.2024 um 22:24 schrieb WM:

Le 05/07/2024 à 15:54, Moebius a écrit :

Am 05.07.2024 um 09:39 schrieb WM:
 > Le 04/07/2024 à 17:25, Moebius a écrit :
 >> Am 04.07.2024 um 17:16 schrieb WM:
 >>
 >>> die Verteilung der ersten Stammbrüche
 >>
 >> es gibt keine "ersten Stammbrüche". Zu _jedem_ Stammbruch gibt es
 >> (abzählbar) unendlich viele kleinere Stammbrüche.
 >
 > Wrong,
>
Nope.

>
Insufficient argument.

>

Hinweis: Wenn s ein Stammbruch ist, dann ist 1/(1/s + 1) ein Stammbruch, der kleiner ist als s.

>
Not true for the smallest unit fraction.

There is NO smallest unit fraction. Dark or not. :^)

Indeed!
Theorem: There is no smallest unit fraction.
Proof: If s is a unit fraction, then there's (by definition) a natural number n such that s = 1/n. Then n + 1 is a natural number to (by one of the peano axioms and the definition of +) and n + 1 > n (by the definition of >). Hence 1/(n + 1) < 1/n (by the rules of the field Q). And hence 1/(1/s + 1) < s (again by the rules of the field Q). Moreover 1/(1/s + 1) is a unit fraction (again by definition). Hence There is a unit fraction, namely 1/(1/s + 1), which is smaller than s.
If we define:
        s is a smallest unit fraction =df s smaller than all other unit fractions
Then the theorem we just proved implies that THERE IS NO "smallest unit fraction".
HENCE in a mathematical context we would not be allowed to talk about "the smallest unit fraction" (as Mückenfuck does) simply because there IS NO such entity.
You see:
The round square is round an square.
The triange with 4 corners has 4 corners.
etc.